{
  "id": "1809.03524",
  "version": "v1",
  "published": "2018-09-10T18:03:24.000Z",
  "updated": "2018-09-10T18:03:24.000Z",
  "title": "Arithmetic representations of fundamental groups II: finiteness",
  "authors": [
    "Daniel Litt"
  ],
  "comment": "30 pages, comments welcome!",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $X$ be a smooth curve over a finitely generated field $k$, and let $\\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod $\\ell$ representations of the geometric fundamental group of $X$. Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero. For example, we show that if $X$ is a normal, connected variety over $\\mathbb{C}$, the (typically infinite) set of representations of $\\pi_1(X^{\\text{an}})$ into $GL_n(\\overline{\\mathbb{Q}_\\ell})$, which come from geometry, has no limit points. As a corollary, we deduce that if $L$ is a finite extension of $\\mathbb{Q}_\\ell$, then the set of representations of $\\pi_1(X^{\\text{an}})$ into $GL_n(L)$, which arise from geometry, is finite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-10T18:03:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic representations",
      "function fields",
      "geometric fundamental group",
      "fontaine-mazur finiteness conjectures",
      "finite extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}